From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 202

of the underlying medium. This becomes apparent when the medium is anisotropic
and has a complex microstructure, which is the case in cerebral white matter where
numerous fiber bundles criss-cross at a resolution much finer than that of dMRI.

6.4 From Diffusion MRI to Tissue Microstructure

Although Stejskal in Eq. ( 6.13 ) formulated the signal for anisotropic diffusion using
a diffusion tensor (DT), the reason he fell short of providing a method for estimating
the DT is perhaps because he was involved with dNMR. In such experiments it was
generally possible to re-orient the experimental setup to align the primary anisotropy
direction with the laboratory frame, sufficing it to measure the diffusion coefficient
in only three directions [ 34 ].
This however changed with MRI and dMRI, when large anisotropic specimen
that couldn't be rotated in the scanner began to be imaged. Imaging such specimen,
e.g. cerebral white matter tissue, or the entire brain, revealed that the diffusion
coefficient measured in such specimen depended upon the direction of the diffusion
encoding gradient. In other words the dMRI signal decay was different along
different gradient directions; or again such dMRI images revealed different contrasts
as the diffusion encoding gradient was rotated. These dMRI images were called
Diffusion Weighted Images (DWIs). DWIs were at first cryptic because while they
revealed the diffusion coefficient, they clearly also indicated that the underlying
tissue was highly anisotropic, but they did not provide a method for inferring
the preferential directions of this anisotropy. The diffusion coefficients com-
puted from these DWIs using Stejskal-Tanner's isotropic signal decay formulation
S (Eq. 6.12 ) were called the apparent diffusion coefficient (ADC),

since these changed in the highly anisotropic tissue depending on the direction of the
diffusion encoding gradient (Fig. 6.5 ). This limitation of the DWI's, and of Stejskal-
Tanner's isotropic signal formulation, shifted the interest from measuring only the
diffusion coefficient to inferring the preferential diffusion anisotropy directions,
or to using diffusion as a probe to infer the tissue's microstructure. This brought
forth a whole new meaning to Stejskal's DT formulation, and it's measurement
from dMRI, since its diagonalisation provided a local coordinate system that was
a good indicator of the preferential diffusion anisotropy directions or the underlying
medium's microstructure.

= S 0 exp( − bD )

6.4.1 Diffusion Tensor Imaging: The Simplest Model

Diffusion tensor imaging (DTI) was introduced by Basser et al. [ 5 , 6 ] in 1994,
which for the first time provided a method for measuring the DT from dMRI and
for inferring the local tissue microstructure from the DT. Starting from Stejskal's
equation, Basser et al. defined the b -matrix, which also accounted for the imaging